#2542923

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2026 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,7)(4,5)(6,8)(10,11)(12,16)(17,18,22,24)(19,21,23,20)(26,29)(27,31);(1,2,5,4)(3,6,8,7)(9,10,13,11)(12,14,16,15)(17,19,22,23)(18,20,24,21)(27,31)(30,32);(2,4)(6,7)(9,10,13,11)(12,15,16,14)(17,20,22,21)(18,23,24,19)(25,26)(27,30)(28,29)(31,32);(1,3,5,8)(2,6,4,7)(9,11,13,10)(12,15,16,14)(17,21,22,20)(18,23,24,19)(25,27)(26,30)(28,31)(29,32);(1,4,5,2)(3,7,8,6)(9,12)(10,14)(11,15)(13,16)(17,18)(19,20)(21,23)(22,24)(25,26)(27,30)(28,29)(31,32)

Polycyclic presentation:

2^9|a^2=gi,b^2=f,c^2=fghi,d^2=f,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2450119);